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Given a double-indexed real sequence $\{ x_{n,m}\}$, do we have

$$ \limsup_{n} \sum_m x_{n,m} \leq \sum_m \limsup_{n} \, x_{n,m}$$

$$ \liminf_{n} \sum_m x_{n,m} \geq \sum_m \liminf_{n} \,x_{n,m}?$$

I am not sure about these, and just have some guess based on how sup and sum commute.

Thanks in advance!

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If the xs are nonnegative, the sum of the liminfs is at most the liminf of the sums. This is Fatou's lemma, see here: en.wikipedia.org/wiki/Fatou%27s_lemma which explains some weaker hypothesis and the corresponding statement for limsups. –  Did Aug 25 '11 at 14:51
    
What do we do if the series $\sum_{m=0}^{\infty}x_{n,m}$ is not convergent? (if the $x_{n,m}$ are not necessary nonnegative, the sequence $\{\sum_{m=0}^Nx_{n,m}\}$ may have no limit at all, even in $\overline{\mathbb R}$). –  Davide Giraudo Aug 25 '11 at 15:15
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1 Answer 1

up vote 3 down vote accepted

In the case where the sequences are nonegative, this is a consequence of the more general Fatou's Lemma.

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Careful (with that axe) Eric... Fatou does not like real valued sequences without some restriction. –  Did Aug 25 '11 at 14:53
    
@Didier: Thanks, you are right. Although, shouldn't the inequality still be true provided the sequences converge? Not sure how to prove this though.. –  Eric Naslund Aug 25 '11 at 14:58
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Eric, try x(n,m)=(-1) if m<n and 0 otherwise. –  Did Aug 25 '11 at 15:05
    
@Didier: I don't think both sides converge in that case? –  Eric Naslund Aug 25 '11 at 16:01
    
Eric, right, then x(n,m)=(-2^(m-n)) if m<n and 0 otherwise: lim_n sum_m x(n,m) is lim_n -1+1/2^n = -1 but sum_m lim_n x(n,m) is sum_m 0 = 0. –  Did Aug 25 '11 at 17:17
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